3.2908 \(\int \frac{(c+d x)^3}{\left (a+b (c+d x)^4\right )^2} \, dx\)

Optimal. Leaf size=23 \[ -\frac{1}{4 b d \left (a+b (c+d x)^4\right )} \]

[Out]

-1/(4*b*d*(a + b*(c + d*x)^4))

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Rubi [A]  time = 0.0206373, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{1}{4 b d \left (a+b (c+d x)^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^3/(a + b*(c + d*x)^4)^2,x]

[Out]

-1/(4*b*d*(a + b*(c + d*x)^4))

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Rubi in Sympy [A]  time = 3.90277, size = 17, normalized size = 0.74 \[ - \frac{1}{4 b d \left (a + b \left (c + d x\right )^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**3/(a+b*(d*x+c)**4)**2,x)

[Out]

-1/(4*b*d*(a + b*(c + d*x)**4))

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Mathematica [A]  time = 0.0188896, size = 23, normalized size = 1. \[ -\frac{1}{4 b d \left (a+b (c+d x)^4\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^3/(a + b*(c + d*x)^4)^2,x]

[Out]

-1/(4*b*d*(a + b*(c + d*x)^4))

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Maple [B]  time = 0., size = 56, normalized size = 2.4 \[ -{\frac{1}{4\,bd \left ( b{d}^{4}{x}^{4}+4\,bc{d}^{3}{x}^{3}+6\,b{c}^{2}{d}^{2}{x}^{2}+4\,b{c}^{3}dx+b{c}^{4}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/(a+b*(d*x+c)^4)^2,x)

[Out]

-1/4/b/d/(b*d^4*x^4+4*b*c*d^3*x^3+6*b*c^2*d^2*x^2+4*b*c^3*d*x+b*c^4+a)

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Maxima [A]  time = 1.39415, size = 28, normalized size = 1.22 \[ -\frac{1}{4 \,{\left ({\left (d x + c\right )}^{4} b + a\right )} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((d*x + c)^4*b + a)^2,x, algorithm="maxima")

[Out]

-1/4/(((d*x + c)^4*b + a)*b*d)

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Fricas [A]  time = 0.208661, size = 89, normalized size = 3.87 \[ -\frac{1}{4 \,{\left (b^{2} d^{5} x^{4} + 4 \, b^{2} c d^{4} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c^{3} d^{2} x +{\left (b^{2} c^{4} + a b\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((d*x + c)^4*b + a)^2,x, algorithm="fricas")

[Out]

-1/4/(b^2*d^5*x^4 + 4*b^2*c*d^4*x^3 + 6*b^2*c^2*d^3*x^2 + 4*b^2*c^3*d^2*x + (b^2
*c^4 + a*b)*d)

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Sympy [A]  time = 22.2603, size = 73, normalized size = 3.17 \[ - \frac{1}{4 a b d + 4 b^{2} c^{4} d + 16 b^{2} c^{3} d^{2} x + 24 b^{2} c^{2} d^{3} x^{2} + 16 b^{2} c d^{4} x^{3} + 4 b^{2} d^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**3/(a+b*(d*x+c)**4)**2,x)

[Out]

-1/(4*a*b*d + 4*b**2*c**4*d + 16*b**2*c**3*d**2*x + 24*b**2*c**2*d**3*x**2 + 16*
b**2*c*d**4*x**3 + 4*b**2*d**5*x**4)

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GIAC/XCAS [A]  time = 0.213658, size = 28, normalized size = 1.22 \[ -\frac{1}{4 \,{\left ({\left (d x + c\right )}^{4} b + a\right )} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((d*x + c)^4*b + a)^2,x, algorithm="giac")

[Out]

-1/4/(((d*x + c)^4*b + a)*b*d)